Abstract
The main intent of this article is to innovate a new iterative method to approximate fixed points of contraction and nonexpansive mappings. We prove that the new iterative method is stable for contraction and has a better rate of convergence than some distinctive iterative methods. Furthermore, some convergence results are proved for nonexpansive mappings. Finally, the solution of a nonlinear fractional difference equation is approximated via the proposed iterative method. Some numerical examples are constructed to support the analytical results and to illustrate the efficiency of the proposed iterative method.
1. Introduction
The fixed point theory is an imperative arm of mathematics. It has become not only a field with significant advancement but also a vital tool for solving different kinds of problems in several fields of mathematics. The fixed point theory is a powerful tool because it has a variety of applications in different fields like differential and integral equations, variational inequalities, approximation theory, economics, biological sciences, medical sciences, engineering, optimization theory, fractal theory, game theory, and control theory. Indeed, the strength of fixed point theory lies in its wide range of applications inside and beyond mathematics.
All over this paper, we presume that is the set of all non-negative integers, a Banach space, , and . A mapping is said to be a contraction mapping if , such that :
If , then is a nonexpansive mapping on .
In 1965, an underlying existence result for fixed points of nonexpansive mappings was proved independently by Browder [1], Göhde [2], and Kirk [3]. After that, fixed point theory for nonexpansive and allied classes of mappings has been examined broadly and has provoked a parallel study in Banach space geometry.
On the other hand, it is well known that the Picard iterative method failed to estimate the fixed points of nonexpansive mappings. So, in 1953, Mann [4] introduced a one-step iterative method to estimate the fixed points of nonexpansive mappings. But, it can be effortlessly seen that Mann’s iterative method was unsuccessful to estimate the fixed points of pseudo-contractive mappings. Therefore, in 1974, Ishikawa [5] defined a two-step Mann iterative method to estimate the fixed points of such type mappings. The speed of convergence of iterative sequence is important from the practical point of view. The faster iterative method saves time while approximating fixed points of nonlinear mappings. Recently, Ali et al. [6], and Garodia and Uddin [7] introduced new iterative methods to achieve a better rate of convergence. Quite recently, several Man-type and Ishikawa-type iterative methods have been studied by different researchers for the approximation of fixed points of single-valued nonlinear functions, e.g., see [8–10]. The following iterative methods are generated by an initial point , where is a self-mapping on .
The S-iterative method (Agarwal et al. [11]) is as follows:
The Picard-S iterative method (Gursoy and Karakaya [12]) can be given as
The Vatan iterative method (Karakaya et al. [13]) can be given as
The Thakur-New iterative method (Thakur et al. [14]) can be given as
The iterative method (Ullah and Arshad [15]) can be given as
The M iterative method (Ullah and Arshad [16]) can be given as
Strongly inspired by the above equations, we define the following a novel iterative method:where and are control sequences in (0, 1).
Remark 1. This iterative method (8) is different from all the iterative methods existing in the literature.
The rest of the manuscript is unified as follows: Section 2 contains some definitions and lemmas that will be used in the main findings. In Section 3, the rate of convergence of the proposed iterative method is compared with some known and remarkable iterative methods by analytically and numerically. The stability of the new iterative method is also discussed with respect to contraction. In Section 4, weak and strong convergence theorems are proven for nonexpansive mappings via the newly defined method. In Section 5, the solution of a nonlinear fractional difference equation is estimated via the proposed iterative method. The conclusion of the paper is given in the last section.
2. Preliminaries
For the purpose of convenience, we recall the following concepts and results that will be used in the sequel.
Lemma 1 (see [17]). Let and be sequences in that satisfy the following inequality:where , with . If , then .
Definition 1 (see [18]). A Banach space satisfies Opial’s condition if for each , holds true, and with .
Definition 2 (see [19]). Let be a Banach space, and be a function. satisfies condition if is an increasing mapping such that , , , and , .
Lemma 2 (see [20]). Let and be sequences in a uniformly convex Banach space with , ,m and , for some . Then, , where and .
Lemma 3 (see [21]). If is a nonexpansive function, where is a uniformly convex Banach space and is a convex closed subset of . Then, is semiclosed on , where stands for identity map on .
Definition 3 (see [22]). Let be a normed space and be a self-mapping on and let two iterative methods and converge identical to the point . Further, assume that the following error estimates are available:where , such that and as .
Definition 4 (see [22]). Let , such that and as . Ifthen converges to faster than to .
And ifthen and have the equal rate of convergence.
Definition 5 (see [23]). Let be an approximate sequence of . Then, an iterative method is defined asfor some relation , such that as , is called -stable; if , , we get .
3. Rate of Convergence and the Stability of the Proposed Method
The motive of the current section is to demonstrate convergence, rate of convergence, and stability results for contraction in an arbitrary Banach space by the proposed iterative method.
Theorem 1. Let be a Banach space and a nonempty, convex closed subset of and be a contraction mapping. Then, the iterative sequence defined in the method (8) converges to one and only one point of . Moreover, the sequence is stable with respect to contraction mapping.
Proof. Since is a contraction mapping, and , and :By iterative method (8), we getUsing equation (15), we getSince and , so with the fact , we getUsing equation (17), we getInductively, we getSince , it concludes that converges to .
Here, we prove the stability of the method (8). Let be an estimate sequence of in , then the sequence defined by the iterative method (8) is and . Now, we show that .
Let , then by the method (8), we obtainPut and , thenSince , as . Thus, by Lemma 1, , that is, .
Conversely, assume that , then we haveThis shows that . Thus, the iterative method (8) is stable with respect to contraction.
The following theorem shows that the iterative method (8) has the better speed of convergence than the methods (2)–(7) for contraction.
Theorem 2. Let be a Banach space and a nonempty, convex closed subset of , and be a contraction. Assume that the sequence is defined by S (2), by Picard-S (3), by Vatan (4), by Thakur-New (5), by (6), by M (7) and by (8) iterative method. Then, the iterative method (8) converges to a fixed point of faster than the , S, Vatan, Picard-S, Thakur-New and M iterative methods.
Proof. In view of equation (19), we obtainAs proved by Sahu ([24], Theorem 3.6), we getFrom Gursoy and Karakaya’s study [12], we getAs proved by Ullah and Arshad ([25], Theorem 4), we getNow, by iterative method (5), we obtainSince and , so with the fact , we haveBy equation (28), we getThus, using the equation (29), we obtain thatInductively, we getLetBy iterative method (6) and using equation (28), we getNow, using equation (33), we getThus, using equation (34), we obtain thatInductively, we getLetBy iterative method (7) and using equation (28), we getNow, using equation (38), we haveBy equation (39), we obtainInductively, we obtainLetNow,Since and , we have
as . Thus, has a better speed of convergence than andimplies as . Thus, has better speed of convergence than .
Next,Thus, as . Hence, has better speed of convergence than .
Furthermore,AndThus, the sequence has better speed of convergence than the sequences , , and .
The following example supports the above result.
Example 1. Take and . Let be given by , for all . Then, is a contraction mapping and admits an unique fixed point . Here, we choose the control sequences and , for all with the initial guess .
The proposed method (8) has better speed of convergence than S, Picard-S, Vatan, Thakur-New, , and M iterative methods with control sequences , , , and the initial point (Tables 1 and 2 and Figure 1). With the same inputs, we compare the CPU time for distinct iterative methods (Table 2).
3.1. Observations
In the present article, we compare the speed of convergence of different iterative methods only in number of iterations. In Table 1, we observe that proposed method (8) converges to a fixed point in 10 iterations and other methods, as shown in Figure 1, and converges more than 10 iterations. Thus, method (8) converges faster than methods (2)–(7).
We also compare the convergence behavior of the proposed iterative method for different initial points. We noticed that the rate of convergence also depends on the initial points, and one can easily see it in Table 3 and Figure 2.
4. Convergence Theorems
First, we demonstrate the following fruitful lemmas that help us to obtain the sequel.
Lemma 4. Let be a uniformly convex Banach space, be a convex closed subset of , and let be a nonexpansive mapping. If is defined by (8), then exists when .
Proof. As is a nonexpansive mapping, so we get and . By the iterative method (8), we obtain Using equation (49), we haveUsing equation (50), we haveHence, is nonincreasing . Hence, exists.
Lemma 5. Let all the assumptions of Lemma 4 be true. If is defined in the method (8), then if and only if is bounded and .
Proof. Presume that and . Then, exists by Lemma 4 and is bounded. Presume thatBy equation (50) and (52), we obtainAs is a nonexpansive mapping, we getNow,Thus,By (53)–(56) and applying Lemma 2, we getNow, by using equation (57), we obtainThus,Conversely, presume that is bounded and . Let , then we haveThis implies that . Thus, we know that consists of only one element because is uniformly convex; hence, .
Theorem 3. Let all the assumptions of Lemma 4 be true. Presume that contents Opial’s property, then the sequence given in (8) converges to a point of weakly.
Proof. Assume and are two subsequences of such that and as (, stands for weakly convergent), where . In view of Lemma 4, exists and by Lemma 5, . In view of Lemma 3, , i.e., , in the same way .
Now, we prove that . If , then using Opial’s condition, we getA contradiction implies that . Thus, and .
Theorem 4. Let all the assumptions of Lemma 4 be true. Then, defined by (8) goes to an fixed element of if and only if , where .
Proof. One can prove the first part easily. For reverse, presume . In view of Result 4.1, exists, and is given.
We now prove that converges to an element of . Since , for , with .Specifically, . So with Now, for , we haveThis shows that the sequence is Cauchy in , so that an element and . Now, , and thus, .
Theorem 5. Let all the assumptions of Lemma 4 be true and contents condition . Then, defined by (8) converges strongly to a point of .
Proof. Using Lemma 5 and condition , we obtainThis implies thatThus, the result is followed by Theorem 4.
Example 2. Let with the norm , , and . A function is given asThen, is a nonexpansive mapping and has a fixed point , whether is not a contraction.
The proposed method (8) goes to the fixed point of the function better than S, Picard-S, Vatan, Thakur-new, , and M iterative methods with the control sequences , , , and initial point (0.5, 0.8) (Tables 4–6 and Figure 3). With the same inputs, we compare the CPU time for the conbergence of distinct iterative methods (Table 6).
5. Solution of a Nonlinear Fractional Difference Equation
The main intent of the present section is to approximate the solution of a fractional difference equation via an iterative method (8). Consider the following:where indicates the Caputo-like discrete fractional difference of order , is a continuous function, and is a real Banach space with the norm
It is shown in [26] that is a solution of the initial value problem (IVP) (67) if and only if is a solution of the following relation:
The following lemma plays a key part to demonstrate the main finding of this segment.
Lemma 6 (see [26]). We haveNow, set and is a real Banach space, where . Presume that the following assumptions are true.
: Assume is a locally Lipschitz continuous function with constant , ,:The existence and uniqueness of a solution of the IVP (67) can be found in [26].
Presently, we are going to demonstrate the main result of the present section.
Theorem 6. Presume that the conditions and are satisfied. Let an operator be given byfor . Then, the sequence developed by iterative method (8) converges to a unique solution of IVP (67).
Proof. It is enough to show the operator is a contraction mapping on . For any , we obtain thatThus, is a contraction on . Hence, by applying Theorem 1, the sequence converges to a unique solution of IVP (67).
6. Conclusion
The main intent of this article was to propose a novel and effective iterative method for the estimation of fixed points of contraction and nonexpansive mappings in the frame work of Banach space. It is also shown that the new iterative method is stable with respect to contraction mapping. The rate of convergence of the distinct iterative methods has been discussed. Therefore, the newly introduced iterative method is more efficient and effective than the previously defined iterative methods. The researchers may apply the new iterative method to approximate the solution of nonlinear problems to achieve a better rate of convergence. Besides, the results of the present paper generalize and amplify the relevant results in the literature. Thus, it can be concluded that the work done in the paper is new and useful in the area of nonlinear analysis. The fixed point iterative methods are very useful to estimate the solution of nonlinear differential equations, nonlinear fractional differential equations, nonlinear integrodifferential equations, etc. The interested researchers may apply the proposed iterative method to estimate the solution of above discussed problems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.